The proofs of Corollary 1.7 is fine. I had misunderstood his proof. Le me just repeat de Shalit's argument.
(i) We have $[F[E[{\frak{m}}]]:F]\leq \# \mathcal{O}_{K}/\frak{m}$ and $K({\frak{m}\frak{g}})=F[E[{{\frak{m}}}]]\cdot K(\frak{g})$
The key observation is that the set of roots of unity of $F[\frak{g}]$ which are congruent to $1$ modulo $\frak{g}$ is reduced to $\{1\}$ (this comes from our assumption that the conductor of the Groessencharacter associated to $E/F$ divides $\frak{g}$). From the previous observation, it follows from class field theory that
(ii) $[ K( {\frak{m}} {\frak{g}} ) : K ({\frak{g}}) ]=\# \mathcal{O}_{K}/\frak{m}$.
(iii) The result now follows by combining (i), (ii) and the Hasse diagram which appears below Corollary 1.7.